The nonclassical group analysis of the heat equation

The nonclassical method of reduction was devised originally by Bluman and Cole in 1969, to find new exact solutions of the heat equation. Much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance. However, the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved, although particular solutions have been given. Recently, Arrigo, Goard, and Broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical Lie symmetry reduction solutions. Their arguments leave open the problem of what is the general nonclassical group action, and its effect on the relevant solution of the heat equation. In this article, both these problems are solved. In the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of Burgers' equation. (C) 1999 Academic Press.